↑

Properties

Label 2-891-99.25-c1-0-18
Degree $2$
Conductor $891$
Sign $-0.541 - 0.840i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.169 + 1.60i)2-s + (−0.604 + 0.128i)4-s + (−0.0399 + 0.379i)5-s + (2.00 + 2.22i)7-s + (0.690 + 2.12i)8-s − 0.618·10-s + (−0.122 − 3.31i)11-s + (5.69 + 2.53i)13-s + (−3.24 + 3.60i)14-s + (−4.43 + 1.97i)16-s + (−0.5 − 0.363i)17-s + (−0.263 − 0.812i)19-s + (−0.0246 − 0.234i)20-s + (5.31 − 0.757i)22-s + (−2.73 − 4.73i)23-s + ⋯
L(s)  = 1  + (0.119 + 1.13i)2-s + (−0.302 + 0.0642i)4-s + (−0.0178 + 0.169i)5-s + (0.758 + 0.842i)7-s + (0.244 + 0.751i)8-s − 0.195·10-s + (−0.0369 − 0.999i)11-s + (1.58 + 0.703i)13-s + (−0.868 + 0.964i)14-s + (−1.10 + 0.493i)16-s + (−0.121 − 0.0881i)17-s + (−0.0605 − 0.186i)19-s + (−0.00551 − 0.0524i)20-s + (1.13 − 0.161i)22-s + (−0.570 − 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983104 + 1.80150i\)
\(L(\frac12)\) \(\approx\) \(0.983104 + 1.80150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (0.122 + 3.31i)T \)
good2 \( 1 + (-0.169 - 1.60i)T + (-1.95 + 0.415i)T^{2} \)
5 \( 1 + (0.0399 - 0.379i)T + (-4.89 - 1.03i)T^{2} \)
7 \( 1 + (-2.00 - 2.22i)T + (-0.731 + 6.96i)T^{2} \)
13 \( 1 + (-5.69 - 2.53i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.73 + 4.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.99 - 3.32i)T + (-3.03 + 28.8i)T^{2} \)
31 \( 1 + (3.52 + 1.56i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (1.30 - 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.97 - 4.41i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (0.881 - 1.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.604 + 0.128i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (5.97 - 4.33i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.20 - 1.10i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-1.04 + 0.466i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (5.28 + 9.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.7 - 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.381 + 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.0551 + 0.524i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-11.6 + 5.16i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 + (1.57 + 14.9i)T + (-94.8 + 20.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.66188774873913885718431736936, −9.054158048451258159002236943405, −8.523131864225256639009310791475, −8.016507558482450210910902588512, −6.71358106879699789799664625520, −6.24343528988620535263346538885, −5.38870990973462864124764959230, −4.48548908561714217573946919297, −3.05928456405936858862473901979, −1.67558555799440511201706063854, 1.07303703572267801907247879913, 1.98508871346373028724761419343, 3.44357542081351766963134011926, 4.11978704872889905410302339495, 5.12432533717682614774387799701, 6.43226868782632969587339026940, 7.39350556890794203498092781542, 8.174170248839187805814819923204, 9.227321687701816844915502522866, 10.22610463744899721705846826478

Graph of the $Z$-function along the critical line